{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 0 1 93 41 59 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 
-1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 
93 41 59 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 93 41 59 0 0 
1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 93 41 59 0 0 1 0 0 0 0 0 
0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 
0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 257 13 "Least Squares" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 256 42 "Part 1.  An example: Data on cancer deaths" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "A
nswer the questions in Part 1 here." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "Part 2.  Modeling the cancer d
eath data" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 18 "Enter the vectors " }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT 
-1 2 ", " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 6 ", and " }{XPPEDIT 
18 0 "y" "6#%\"yG" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "with(linalg):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "one:=vector([
1,1,1,1,1,1,1,1,1]); x:=vector([2.5,2.6,3.4,1.3,1.6,3.8,11.6,6.4,8.3])
; y:=vector([147,130,130,114,138,162,208,178,210]);\n" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 19 "Construct a matrix " }{XPPEDIT 18 0 "U" "
6#%\"UG" }{TEXT -1 49 " whose columns are an orthonormal basis for Spa
n(" }{XPPEDIT 18 0 "1" "6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x" 
"6#%\"xG" }{TEXT -1 2 ")." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Z:=tr
anspose(GramSchmidt([one,x])); z1:=col(Z,1): z2:=col(Z,2): u1:=z1/norm
(z1,2): u2:=z2/norm(z2,2): U:=augment(u1,u2);\n" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 23 "Find the projection of " }{XPPEDIT 18 0 "y" "6#%\"yG
" }{TEXT -1 28 " on the span of the vectors " }{XPPEDIT 18 0 "1" "6#\"
\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 1 ".
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 29 "Part 3.  The normal equa
tions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 21 "Construct the matrix " }{XPPEDIT 18 0 "X" "6#%\"XG" }
{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "X:=augment(one,x);
\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Find the coefficient matrix
 and constant vector for the normal equations." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Solve the nor
mal equations for coefficients " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 28 " for the least
 squares line " }{XPPEDIT 18 0 "y=m*x+b" "6#/%\"yG,&*&%\"mG\"\"\"%\"xG
F(F(%\"bGF(" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Confirm that your coefficients giv
e the same solution as the projection of " }{XPPEDIT 18 0 "y" "6#%\"yG
" }{TEXT -1 8 " on Col(" }{XPPEDIT 18 0 "X" "6#%\"XG" }{TEXT -1 2 ").
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 16 "Part 4.  Summary" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "13" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }